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A Comparison of Matrix-Free Isogeometric Galerkin and Collocation Methods for Karhunen–Loève Expansion

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Abstract

Numerical computation of the Karhunen–Loève expansion is computationally challenging in terms of both memory requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the K–L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed by us in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2]. Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth covariance kernels, while the collocation method performs slightly better for rough covariance kernels.

Michal, René and I are part of the younger generation of researchers in Hannover. As a junior faculty member, my vision for the future is borne by the legacy of excellent research, teaching and mentorship that Peter Wriggers and the many colleagues he attracted to Hannover have maintained over the past 25 years. Michal is one of the many outstanding graduates of the educational programs in computational methods initiated by Peter. It was also Peter who established the contact between us at UT Austin, where the foundations for this work were laid. (Dominik Schillinger).

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Notes

  1. 1.

    Formation and assembly costs for a standard Galerkin method scale \(\mathcal {O}(N^2_e (p+1)^{3d}))\), where \(N_e\) is the number of finite elements, p is the polynomial degree and d is the spatial dimension.

  2. 2.

    In general the stability and convergence analysis are challenging in the context of collocation methods.

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Mika, M.L., Hiemstra, R.R., Schillinger, D., Hughes, T.J.R. (2022). A Comparison of Matrix-Free Isogeometric Galerkin and Collocation Methods for Karhunen–Loève Expansion. In: Aldakheel, F., Hudobivnik, B., Soleimani, M., Wessels, H., Weißenfels, C., Marino, M. (eds) Current Trends and Open Problems in Computational Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-87312-7_32

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  • DOI: https://doi.org/10.1007/978-3-030-87312-7_32

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